IODS course project
Marc Denojean-Mairet
- Introduction to Open Data Science - Course Project
- Logistic regression
- Chapter 4: Clustering and Classification
- Part 4: standardize the dataset
- Part 5: linear discriminant analysis
- Part 6: Save the crime categories
- Part 7: Reload the Boston dataset and standardize the dataset
- Chapter 5: Dimensionality reduction techniques
- Part 1: Show a graphical overview of the data and show summaries of the variables in the data
- Part 2: Perform principal component analysis (PCA) on the not standardized human data.
- Part 3: Standardize the variables in the human data and repeat the above analysis.
- Part 4: personal interpretations of the first two principal component dimensions based on the biplot drawn after PCA on the standardized human data
- Part 5: Load the tea dataset from the package Factominer
- Chapter 6: Analysis of longitudinal data
Introduction to Open Data Science - Course Project
# This is a so-called "R chunk" where you can write R code.
date()## [1] "Sun Dec 12 23:05:33 2021"I am interested in this course because of my research and I am expecting that this course will be a strong introduction to data science. The course should give me enough knowledge to be able to use data science process in my research.
I learned about the course via an email sent to me by my department.
My GitHub repository can be found here.
Here is my course diary web page.
For this analysis, a data frame named learningAnalysis2014 is created. The CSV file named “learning2014.csv” is read. The data frame consist of 7 variables (gender ,Age ,attitude, deep, stra, surf, Points) and 166 observations. The data are from a survey of statistics students. The data include the global attitude of the students toward statistics and their exam points. deep”, “stra” and “surf” are combined variables by taking the mean. “attitude” was scaled based on Likert scale (1-5) by dividing the “Attitude” column by 10.
More information about the data can be found here (https://www.mv.helsinki.fi/home/kvehkala/JYTmooc/JYTOPKYS3-meta.txt)
learningAnalysis2014 <- read.csv(file = 'data/learning2014.csv')
dim(learningAnalysis2014)## [1] 166 7str(learningAnalysis2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : chr "F" "M" "F" "M" ...
## $ Age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ Points : int 25 12 24 10 22 21 21 31 24 26 ...These are plots of all the relationship among the variables. From the visualization, we see some positive and negative correlations. An interesting correlation is attitude and points. As expected, we see a negative correlation with surf and deep. The most negative correlation is with deep and points. We also see that there are more female than male students. However, from the plots, there is no good fit between gender and points and there is no strong correlations.
The summary table gives us more information about the means of the variables.
pairs(learningAnalysis2014[-1], col = "red")
library(ggplot2)
library(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2ggpairs(learningAnalysis2014, mapping = aes(col = gender, alpha = 0.3), lower = list(combo = wrap("facethist", bins = 20)))
summary(learningAnalysis2014)## gender Age attitude deep
## Length:166 Min. :17.00 Min. :1.400 Min. :1.583
## Class :character 1st Qu.:21.00 1st Qu.:2.600 1st Qu.:3.333
## Mode :character Median :22.00 Median :3.200 Median :3.667
## Mean :25.51 Mean :3.143 Mean :3.680
## 3rd Qu.:27.00 3rd Qu.:3.700 3rd Qu.:4.083
## Max. :55.00 Max. :5.000 Max. :4.917
## stra surf Points
## Min. :1.250 Min. :1.583 Min. : 7.00
## 1st Qu.:2.625 1st Qu.:2.417 1st Qu.:19.00
## Median :3.188 Median :2.833 Median :23.00
## Mean :3.121 Mean :2.787 Mean :22.72
## 3rd Qu.:3.625 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :5.000 Max. :4.333 Max. :33.00I am using the following 3 variables to explain Points: ‘attitude’, ‘stra’, and ‘deep’.
ggpairs(learningAnalysis2014, lower = list(combo = wrap("facethist", bins = 20)))
my_model <- lm(Points ~ attitude + stra + deep, data = learningAnalysis2014)
summary(my_model)##
## Call:
## lm(formula = Points ~ attitude + stra + deep, data = learningAnalysis2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.5239 -3.4276 0.5474 3.8220 11.5112
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.3915 3.4077 3.343 0.00103 **
## attitude 3.5254 0.5683 6.203 4.44e-09 ***
## stra 0.9621 0.5367 1.793 0.07489 .
## deep -0.7492 0.7507 -0.998 0.31974
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.289 on 162 degrees of freedom
## Multiple R-squared: 0.2097, Adjusted R-squared: 0.195
## F-statistic: 14.33 on 3 and 162 DF, p-value: 2.521e-08From the above summary table, we see that median for the residual is 0.5474. This would suggest that it will be difficult to predict points based on attitude, stra, and deep. However, it is very difficult to predict human behaviors and 0.5 residual could be acceptable in this case.
From the coefficients table, we see that the p-value for deep is high which would mean that deep does not affect much points. stra and attitude are better to predict points.
Below is a new regression where I have removed deep.
my_model2 <- lm(Points ~ attitude + stra, data = learningAnalysis2014)
summary(my_model2)##
## Call:
## lm(formula = Points ~ attitude + stra, data = learningAnalysis2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.6436 -3.3113 0.5575 3.7928 10.9295
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.9729 2.3959 3.745 0.00025 ***
## attitude 3.4658 0.5652 6.132 6.31e-09 ***
## stra 0.9137 0.5345 1.709 0.08927 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.289 on 163 degrees of freedom
## Multiple R-squared: 0.2048, Adjusted R-squared: 0.1951
## F-statistic: 20.99 on 2 and 163 DF, p-value: 7.734e-09Based on the above summary table, we see that by removing deep, the fit was not improved and actually got worse.
Below are diagnostic plots function: Residuals vs Fitted values, Normal QQ-plot and Residuals vs Leverage. The QQ-plot shows a reasonable fit which shows good ‘normality’. The Residuals vs Fitted values plot shows that it is reasonable since it shows randomness. The Residuals vs Leverage plot shows regular error which would imply regular leverage.
my_model2 <- lm(Points ~ attitude + stra, data = learningAnalysis2014)
par(mfrow = c(2,2))
plot(my_model2, which = c(1,2,5))
Logistic regression
Read new data frame named alc.csv to studentAlc and print variable names.
#studentAlc <- read.table("~/IODS-project/IODS-project/data/pormath.csv", sep = ",", header = TRUE)
studentAlc <- read.table("~/IODS-project/IODS-project/data/alc.csv", sep = ",", header = TRUE)
dim(studentAlc)## [1] 370 35colnames(studentAlc)## [1] "school" "sex" "age" "address" "famsize"
## [6] "Pstatus" "Medu" "Fedu" "Mjob" "Fjob"
## [11] "reason" "guardian" "traveltime" "studytime" "schoolsup"
## [16] "famsup" "activities" "nursery" "higher" "internet"
## [21] "romantic" "famrel" "freetime" "goout" "Dalc"
## [26] "Walc" "health" "failures" "paid" "absences"
## [31] "G1" "G2" "G3" "alc_use" "high_use"Data Set Information:
“This data approach student achievement in secondary education of two Portuguese schools. The data attributes include student grades, demographic, social and school related features) and it was collected by using school reports and questionnaires. Two datasets are provided regarding the performance in two distinct subjects: Mathematics (mat) and Portuguese language (por). In [Cortez and Silva, 2008], the two datasets were modeled under binary/five-level classification and regression tasks. Important note: the target attribute G3 has a strong correlation with attributes G2 and G1. This occurs because G3 is the final year grade (issued at the 3rd period), while G1 and G2 correspond to the 1st and 2nd period grades. It is more difficult to predict G3 without G2 and G1, but such prediction is much more useful (see paper source for more details).”
The above information is from UCI Machine Learning Repository.
More information about the data sets can be found here:(https://archive.ics.uci.edu/ml/datasets/Student+Performance)
I chose the following 4 variables to predict high use of alcohol: G3, absences, Pstatus, and health. I have chosen these variables thinking that they would be easily available to a school without the need to survey students.
G3:
My assumption here is that low grade is an indication of high alcohol use since high use of alcohol can affect cognitive functions such as memory.
absences:
High number of absences could be an indication of high alcohol use as it would affect one’s schedule. Absenses could be due to being sick after consuming a lot of alcohol.
goout
Going out with friend a lot might raise alcohol consumption since there will be more opportunities to consume alcohol.
health:
Poor health could be an indication of high alcohol consumption. Alcohol can negatively affect physical and mental health.
These are plots of all variables
# access the tidyverse libraries tidyr, dplyr, ggplot2
library(tidyr); library(dplyr); library(ggplot2)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union# draw a bar plot of each variable
gather(studentAlc) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar()
Box plots of my chosen variables vs high-use of alcohol
# initialize a plot of high_use and G3
g1 <- ggplot(studentAlc, aes(x = high_use, y = G3, col = sex))
# define the plot as a boxplot and draw it
g1 + geom_boxplot() + ylab("grade") + ggtitle("Student final grade by alcohol consumption and sex")
# initialise a plot of high_use and absences
g2 <- ggplot(studentAlc, aes(x = high_use, y = absences, col = sex))
# define the plot as a boxplot and draw it
g2 + geom_boxplot() + ggtitle("Student absences by alcohol consumption and sex")
# initialize a plot of high_use and
g3 <- ggplot(studentAlc, aes(x = high_use, y = goout, col = sex))
# define the plot as a boxplot and draw it
g3 + geom_boxplot() + ylab("going out") + ggtitle("Student going out with friends by alcohol consumption and sex")
g4 <- ggplot(studentAlc, aes(x = high_use, y = health, col = sex))
# define the plot as a boxplot and draw it
g4 + geom_boxplot() + ylab("health") + ggtitle("Student health by alcohol consumption and sex")
Numerical exploration of my chosen variables
# produce summary statistics by grade
studentAlc %>% group_by(sex, high_use) %>% summarise(count = n(), mean_grade = mean(G3))## `summarise()` has grouped output by 'sex'. You can override using the `.groups` argument.## # A tibble: 4 x 4
## # Groups: sex [2]
## sex high_use count mean_grade
## <chr> <lgl> <int> <dbl>
## 1 F FALSE 154 11.4
## 2 F TRUE 41 11.8
## 3 M FALSE 105 12.3
## 4 M TRUE 70 10.3# produce summary statistics by absences
studentAlc %>% group_by(sex, high_use) %>% summarise(count = n(), mean_absences = mean(absences))## `summarise()` has grouped output by 'sex'. You can override using the `.groups` argument.## # A tibble: 4 x 4
## # Groups: sex [2]
## sex high_use count mean_absences
## <chr> <lgl> <int> <dbl>
## 1 F FALSE 154 4.25
## 2 F TRUE 41 6.85
## 3 M FALSE 105 2.91
## 4 M TRUE 70 6.1# produce summary statistics by going out
studentAlc %>% group_by(sex, high_use) %>% summarise(count = n(), mean_going_out = mean(goout))## `summarise()` has grouped output by 'sex'. You can override using the `.groups` argument.## # A tibble: 4 x 4
## # Groups: sex [2]
## sex high_use count mean_going_out
## <chr> <lgl> <int> <dbl>
## 1 F FALSE 154 2.95
## 2 F TRUE 41 3.39
## 3 M FALSE 105 2.70
## 4 M TRUE 70 3.93# produce summary statistics by health
studentAlc %>% group_by(sex, high_use) %>% summarise(count = n(), mean_health = mean(health))## `summarise()` has grouped output by 'sex'. You can override using the `.groups` argument.## # A tibble: 4 x 4
## # Groups: sex [2]
## sex high_use count mean_health
## <chr> <lgl> <int> <dbl>
## 1 F FALSE 154 3.37
## 2 F TRUE 41 3.39
## 3 M FALSE 105 3.67
## 4 M TRUE 70 3.93Interpretation of the above plots and tables
We see that grades are lower for male when high_use of alcohol is true. Female seems to be less negatively affected. Final grade is not as a strong predictor as I thought. Absences and high_use of alcohol seems to have a good correlation. Female seems to have a little more absences than male when high_use of alcohol is true. going out shows some correlation for male and female for high_use of alcohol. The more a student go out the more he or she consume alcohol. Health does not seem to have a strong correlation with high_use of alcohol. I would have thought that there would have been a stronger correlation. It is possible that the participants did not truthfully answer this question or that participants are not aware of their overall health (mental and physical). This seems to be especially true for male. Female have a broader range of answers with 25% of them describing their health below 2 (Q1 high_use = true).
Logistic regression to statistically explore the relationship between my chosen variables
For this model, high_use is the target variable and final grade, absences, going out, and health are the predictors. I did not include sex because if a school needs to predict high_use of alcohol, sex might add an unnecessary bias. For instance, male students might be watched more carefully than female students because male seems to have higher high_use of alcohol.
# find the model with glm()
m <- glm(high_use ~ G3 + absences + goout + health, data = studentAlc, family = "binomial")
# print out a summary of the model
summary(m)##
## Call:
## glm(formula = high_use ~ G3 + absences + goout + health, family = "binomial",
## data = studentAlc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.8342 -0.7505 -0.5508 0.9357 2.3172
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -3.70363 0.79076 -4.684 2.82e-06 ***
## G3 -0.03852 0.03935 -0.979 0.327587
## absences 0.07436 0.02212 3.362 0.000773 ***
## goout 0.72459 0.11941 6.068 1.29e-09 ***
## health 0.15179 0.09209 1.648 0.099275 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 452.04 on 369 degrees of freedom
## Residual deviance: 386.07 on 365 degrees of freedom
## AIC: 396.07
##
## Number of Fisher Scoring iterations: 4# print out the coefficients of the model
coef(m)## (Intercept) G3 absences goout health
## -3.70363174 -0.03852050 0.07435996 0.72459398 0.15179099# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp## Waiting for profiling to be done...# print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.0246339 0.004960425 0.1109072
## G3 0.9622120 0.890726272 1.0397748
## absences 1.0771945 1.033123265 1.1280521
## goout 2.0638929 1.642693029 2.6260422
## health 1.1639169 0.973629481 1.3981834Interpretation of the above model
From the model P-values, we see that final absences and going out (goout) are likely relevant variables to explain high_use of alcohol. However, final grade and health show that the data are providing little evidence that these variables are needed to explain high_use.
From the Odd Ratio table, we see that absences, going out, and health have an OR greater than 1. This would imply that these variables are positively associated with high_use of alcohol. Going out has a Odd Ratio of 2 showing a very strong positive association with high_use of alcohol. Final grade is close to 1 as well. This would imply that the positive association is not as strong as the other variables.
It seems that most of my chosen variables have some positive association with high_use. Therefore, they could be used to correctly predict high_use.
Predictive power of my model
# predict() the probability of high_use
probabilities <- predict(m, type = "response")
# add the predicted probabilities to 'studentAlc'
studentAlc <- mutate(studentAlc, probability = probabilities)
# use the probabilities to make a prediction of high_use
studentAlc <- mutate(studentAlc, prediction = probability > 0.5)
# see the last ten original classes, predicted probabilities, and class predictions
select(studentAlc, G3, absences, goout, health, high_use, probability, prediction) %>% tail(10)## G3 absences goout health high_use probability prediction
## 361 2 7 3 3 TRUE 0.34728425 FALSE
## 362 11 3 3 3 TRUE 0.21838164 FALSE
## 363 10 2 1 5 TRUE 0.07895958 FALSE
## 364 16 4 4 2 TRUE 0.30564438 FALSE
## 365 12 3 2 3 FALSE 0.11524638 FALSE
## 366 8 4 3 3 TRUE 0.25252304 FALSE
## 367 14 0 2 5 FALSE 0.11559977 FALSE
## 368 9 4 4 5 TRUE 0.47613178 FALSE
## 369 10 8 4 2 TRUE 0.42751451 FALSE
## 370 0 0 2 5 FALSE 0.18309931 FALSE# tabulate the target variable versus the predictions
table(high_use = studentAlc$high_use, prediction = studentAlc$prediction)## prediction
## high_use FALSE TRUE
## FALSE 233 26
## TRUE 65 46# initialize a plot of 'high_use' versus 'probability' in 'studentAlc'
g <- ggplot(studentAlc, aes(x = probability, y = high_use, col = prediction))
# define the geom as points and draw the plot
g + geom_point()
# tabulate the target variable versus the predictions
table(high_use = studentAlc$high_use, prediction = studentAlc$prediction) %>% prop.table %>% addmargins## prediction
## high_use FALSE TRUE Sum
## FALSE 0.62972973 0.07027027 0.70000000
## TRUE 0.17567568 0.12432432 0.30000000
## Sum 0.80540541 0.19459459 1.00000000# define a loss function (average prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = studentAlc$high_use, prob = studentAlc$probability)## [1] 0.2459459The goal of a loss function is to get a small number as possible. The loss function here is around 0.25. Therefore, the model does not predict correctly 25% of the time. The model perform better than the simple guessing strategy.
10-fold cross-validation
Bonus
# K-fold cross-validation
library(boot)
cv <- cv.glm(data = studentAlc, cost = loss_func, glmfit = m, K = 10)
# average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.2567568The result of the cross-validation (K=10) is similar to the previous result of the loss function. My model seems to produce a similar result as the model found in DataCamp.
Super Bonus
Many variables
# New model with many variables
m <- glm(high_use ~ G3 + absences + goout + health + studytime + failures + freetime + famrel, data = studentAlc, family = "binomial")
# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp## Waiting for profiling to be done...# print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.1112482 0.01462431 0.7969864
## G3 1.0004152 0.91884143 1.0909553
## absences 1.0707813 1.02620258 1.1197948
## goout 2.0410506 1.59736372 2.6461037
## health 1.1744770 0.97382116 1.4240680
## studytime 0.6138411 0.43227534 0.8559677
## failures 1.3512750 0.85082981 2.1704783
## freetime 1.1869927 0.89658690 1.5765989
## famrel 0.6655703 0.49995249 0.8817370# predict() the probability of high_use
probabilities <- predict(m, type = "response")
# add the predicted probabilities to 'studentAlc'
studentAlc <- mutate(studentAlc, probability = probabilities)
# use the probabilities to make a prediction of high_use
studentAlc <- mutate(studentAlc, prediction = probability > 0.5)
# define a loss function (average prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = studentAlc$high_use, prob = studentAlc$probability)## [1] 0.2324324# K-fold cross-validation
library(boot)
cv <- cv.glm(data = studentAlc, cost = loss_func, glmfit = m, K = 10)
# average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.2513514New model with 2 variable only
# New model with 2 variables
m <- glm(high_use ~ absences + goout, data = studentAlc, family = "binomial")
# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp## Waiting for profiling to be done...# print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.02644199 0.01082308 0.06045431
## absences 1.07930582 1.03477610 1.13034717
## goout 2.08298255 1.66258410 2.64170298# predict() the probability of high_use
probabilities <- predict(m, type = "response")
# add the predicted probabilities to 'studentAlc'
studentAlc <- mutate(studentAlc, probability = probabilities)
# use the probabilities to make a prediction of high_use
studentAlc <- mutate(studentAlc, prediction = probability > 0.5)
# define a loss function (average prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = studentAlc$high_use, prob = studentAlc$probability)## [1] 0.2378378# K-fold cross-validation
library(boot)
cv <- cv.glm(data = studentAlc, cost = loss_func, glmfit = m, K = 10)
# average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.2405405It seems that the result is not greatly improved when going from many variables to only 2 variables.
Chapter 4: Clustering and Classification
Part 2: explore the structure and the dimensions of the data
Loading the Boston data.
# set plots size
knitr::opts_chunk$set(fig.width=16, fig.height=10)
#code from DataCamp.
#access the MASS package
library (dplyr)
library(MASS)
library(corrplot)
library(tidyr)
#load the data
data("Boston")
#explore the dataset
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...dim(Boston)## [1] 506 14summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Description:
The Boston data set represent the housing values in Suburbs of Boston.
The Boston data frame has 506 rows and 14 columns.
This data frame contains the following columns:
crim = per capita crime rate by town
zn = proportion of residential land zoned for lots over 25,000 sq.ft
indus = proportion of non-retail business acres per town
chas = Charles River dummy variable (= 1 if tract bounds river; 0 otherwise)
nox = nitrogen oxides concentration (parts per 10 million)
rm = average number of rooms per dwelling
age = proportion of owner-occupied units built prior to 1940
dis = weighted mean of distances to five Boston employment centres
rad = index of accessibility to radial highways
tax = full-value property-tax rate per $10,000
ptratio = pupil-teacher ratio by town
black = 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town
lstat = lower status of the population (percent)
medv = median value of owner-occupied homes in $1000s
Source:
Harrison, D. and Rubinfeld, D.L. (1978) Hedonic prices and the demand for clean air. J. Environ. Economics and Management 5, 81–102.
Belsley D.A., Kuh, E. and Welsch, R.E. (1980) Regression Diagnostics. Identifying Influential Data and Sources of Collinearity. New York: Wiley.
Part 3: graphical overview of the data
#plot matrix of the variables
pairs(Boston, gap=1/30)
# MASS, corrplot, tidyr and Boston dataset are available
# calculate the correlation matrix and round it
cor_matrix<-cor(Boston) %>% round(digits = 2)
# print the correlation matrix
cor_matrix## crim zn indus chas nox rm age dis rad tax ptratio
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72 0.38
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04 -0.12
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67 0.19
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29 -0.36
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51 0.26
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53 -0.23
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91 0.46
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00 0.46
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1.00
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54 0.37
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47 -0.51
## black lstat medv
## crim -0.39 0.46 -0.39
## zn 0.18 -0.41 0.36
## indus -0.36 0.60 -0.48
## chas 0.05 -0.05 0.18
## nox -0.38 0.59 -0.43
## rm 0.13 -0.61 0.70
## age -0.27 0.60 -0.38
## dis 0.29 -0.50 0.25
## rad -0.44 0.49 -0.38
## tax -0.44 0.54 -0.47
## ptratio -0.18 0.37 -0.51
## black 1.00 -0.37 0.33
## lstat -0.37 1.00 -0.74
## medv 0.33 -0.74 1.00# visualize the correlation matrix
corrplot(cor_matrix, method="circle", type="upper", cl.pos="b", tl.pos="d", tl.cex = 0.6)
corrplot(cor_matrix, method = 'number', type="upper")
Looking at the data, we see that medv has a positive correlation with rm and a negative correlation with lstat. This make sense as a house would have a higher price based on the number of rooms. As well, house located in a lower status population area would have a lower price. We also see that nox has a positive correlation with indus and age. nox has a negative correlation with dis. The higher the number of industry, the higher the emission of nitrogen oxides. Older houses would probably be mostly in old industrial areas of the city. medv has a positive correlation with crim but only at 0.33. At first glance, the value of houses is mainly due to th e number of rooms per dwelling. Taxes has a small negative correlation with medv.
Part 4: standardize the dataset
During scaling of the data, the mean is subtracted from the column and the difference is divided by the standard deviation. This is an example of the data before and after scaling.
Before:
row 1: rm = 6.575
After:
row 1: rm = 0.413262920
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix" "array"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)Use the quantiles as the break points in the categorical variable and divide the dataset to train and test sets
# summary of the scaled crime rate
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, labels = c("low", "med_low", "med_high", "high"))
# look at the table of the new factor crime
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)Now, 80% of the data belongs to the train set. Saved the crime categories from the test set and removed the categorical crime variable from the test dataset.
Part 5: linear discriminant analysis
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2574257 0.2524752 0.2475248 0.2425743
##
## Group means:
## zn indus chas nox rm age
## low 0.9204904 -0.91193584 -0.12090214 -0.8782791 0.4716162 -0.8914111
## med_low -0.1749097 -0.22715264 -0.04073494 -0.5424073 -0.2011207 -0.3333607
## med_high -0.3749021 0.08343905 0.20012296 0.3259898 0.1504028 0.4280137
## high -0.4872402 1.01719597 -0.07145661 1.0780091 -0.3975976 0.8214925
## dis rad tax ptratio black lstat
## low 0.8863868 -0.6892330 -0.7626524 -0.46090827 0.37941206 -0.7820678
## med_low 0.3239559 -0.5607666 -0.4664819 -0.01251866 0.35867821 -0.1022016
## med_high -0.3589343 -0.4260132 -0.3572338 -0.23581822 0.06956764 0.0109559
## high -0.8683550 1.6373367 1.5134896 0.77985517 -0.84015445 0.9076136
## medv
## low 0.5525236
## med_low -0.0406631
## med_high 0.1520254
## high -0.7451492
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.09291863 0.69985636 -0.87796705
## indus 0.03053132 -0.12468573 0.41899216
## chas -0.10201193 -0.08250128 0.01816864
## nox 0.39531543 -0.67513639 -1.32834260
## rm -0.12218291 -0.06127500 -0.23682516
## age 0.19751937 -0.46873738 -0.20225462
## dis -0.05492489 -0.21359549 0.20455652
## rad 3.36773523 0.90989444 -0.08874001
## tax 0.02342119 0.03991802 0.61440283
## ptratio 0.09227121 0.01644471 -0.24956937
## black -0.11449254 0.02101592 0.18057073
## lstat 0.23164876 -0.23280037 0.40508204
## medv 0.17679314 -0.30905705 -0.05937904
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9511 0.0350 0.0139# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "orange", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 1)
Part 6: Save the crime categories
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 15 5 3 0
## med_low 11 7 6 0
## med_high 1 4 19 2
## high 0 0 0 29The best predictions are for high. Also, the model does not predict very well med_high. We can see in the LD plot that med_high and med_low are in the same side. The model predicts high correctly because the distance between high and low, med_low, and med_high is large.
Part 7: Reload the Boston dataset and standardize the dataset
# load MASS and Boston
library(MASS)
data('Boston')
# scale data
Boston = as.data.frame(scale(Boston))
# euclidean distance matrix
dist_eu <- dist(Boston)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# manhattan distance matrix
dist_man <- dist(Boston, method = 'manhattan')
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618cluster with centers = 3
# k-means clustering
km <-kmeans(Boston, centers = 3)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
cluster with centers = 1
# k-means clustering
km <-kmeans(Boston, centers = 1)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
cluster with centers = 2
# k-means clustering
km <-kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston[6:10], col = km$cluster)
The cluster with centers = 3 seems to be the best.
Bonus and super bonus
library(ggplot2)
# Boston dataset is available
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(Boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <-kmeans(Boston, centers = 2)
# plot the Boston dataset with clusters
pairs(Boston, col = km$cluster)
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
#install.packages("plotly")
library(plotly)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')Chapter 5: Dimensionality reduction techniques
Part 1: Show a graphical overview of the data and show summaries of the variables in the data
# set plots size
knitr::opts_chunk$set(fig.width=16, fig.height=10)
# uploading dataset from: http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt
human <- read.csv("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/human2.txt", sep = ",", header = TRUE)
# code is from DataCamp.
# library
library(GGally)
library(corrplot)
# visualize the 'human_' variables
ggpairs(human)
# compute the correlation matrix and visualize it with corrplot
cor(human) %>% corrplot
# summary of the data
summary(human)## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :0.1717 Min. :0.1857 Min. : 5.40 Min. :49.00
## 1st Qu.:0.7264 1st Qu.:0.5984 1st Qu.:11.25 1st Qu.:66.30
## Median :0.9375 Median :0.7535 Median :13.50 Median :74.20
## Mean :0.8529 Mean :0.7074 Mean :13.18 Mean :71.65
## 3rd Qu.:0.9968 3rd Qu.:0.8535 3rd Qu.:15.20 3rd Qu.:77.25
## Max. :1.4967 Max. :1.0380 Max. :20.20 Max. :83.50
## GNI Mat.Mor Ado.Birth Parli.F
## Min. : 581 Min. : 1.0 Min. : 0.60 Min. : 0.00
## 1st Qu.: 4198 1st Qu.: 11.5 1st Qu.: 12.65 1st Qu.:12.40
## Median : 12040 Median : 49.0 Median : 33.60 Median :19.30
## Mean : 17628 Mean : 149.1 Mean : 47.16 Mean :20.91
## 3rd Qu.: 24512 3rd Qu.: 190.0 3rd Qu.: 71.95 3rd Qu.:27.95
## Max. :123124 Max. :1100.0 Max. :204.80 Max. :57.50Interpretation:
From the above graphs, we see that we have normal, binomial distributions. We also see some strong negative and positve correlations. For instance, Ado.Birth as a positive correlation with Mat.Mor (0.759). As well, Mat.Mor and Life.Exp have a negative correlation (-0.857). Another interesting correlation is the positive correlation between Life.Exp and Edu.Exp (0.789). There is a positive correlation between GNI and Life.Exp (0.627) but it is lower than I would have expected. Life.Exp does increase in countries with high GNI but not much. There is a negative correlation between Ado.Birth and Edu.EXp. This would suggest that the two variables are moving in opposite direction.
Part 2: Perform principal component analysis (PCA) on the not standardized human data.
# code is from DataCamp
# perform principal component analysis (with the SVD method)
pca_human <- prcomp(human)
# draw a biplot of the principal component representation and the original variables
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("red", "blue"))
Part 3: Standardize the variables in the human data and repeat the above analysis.
# code is from DataCamp
# standardize the variables
human_std <- scale(human)
# print out summaries of the standardized variables
summary(human_std)## Edu2.FM Labo.FM Edu.Exp Life.Exp
## Min. :-2.8189 Min. :-2.6247 Min. :-2.7378 Min. :-2.7188
## 1st Qu.:-0.5233 1st Qu.:-0.5484 1st Qu.:-0.6782 1st Qu.:-0.6425
## Median : 0.3503 Median : 0.2316 Median : 0.1140 Median : 0.3056
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5958 3rd Qu.: 0.7350 3rd Qu.: 0.7126 3rd Qu.: 0.6717
## Max. : 2.6646 Max. : 1.6632 Max. : 2.4730 Max. : 1.4218
## GNI Mat.Mor Ado.Birth Parli.F
## Min. :-0.9193 Min. :-0.6992 Min. :-1.1325 Min. :-1.8203
## 1st Qu.:-0.7243 1st Qu.:-0.6496 1st Qu.:-0.8394 1st Qu.:-0.7409
## Median :-0.3013 Median :-0.4726 Median :-0.3298 Median :-0.1403
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.1932 3rd Qu.: 0.6030 3rd Qu.: 0.6127
## Max. : 5.6890 Max. : 4.4899 Max. : 3.8344 Max. : 3.1850# perform principal component analysis (with the SVD method)
pca_human <- prcomp(human_std)
# draw a biplot of the principal component representation and the original variables
biplot(pca_human, choices = 1:2, cex = c(0.8, 1), col = c("red", "blue"))
# create and print out a summary of pca_human
s <- summary(pca_human)
s## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631 0.45900
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595 0.02634
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069 0.98702
## PC8
## Standard deviation 0.32224
## Proportion of Variance 0.01298
## Cumulative Proportion 1.00000# rounded percentages of variance captured by each PC
pca_pr <- round(100*s$importance[2,], digits = 1)
# print out the percentages of variance
pca_pr## PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
## 53.6 16.2 9.6 7.6 5.5 3.6 2.6 1.3# create object pc_lab to be used as axis labels
pc_lab <- paste0(names(pca_pr), " (", pca_pr, "%)")
# draw a biplot
biplot(pca_human, cex = c(0.8, 1), col = c("red", "blue"), xlab = pc_lab[1], ylab = pc_lab[2])
Interpretation:
From the above tables and graphs, we see that the non-standardized data are difficult to interpret. The biplot graph shows most of the country to be on the top right corner and the arrow to have a 0 length. The standardized PCA biplot shows that most of the countries are in the middle of the graph. We see that arrows of Mat.Mor and Ado.Birth have a small angle (meaning that they are correlated). They have a positive correlation and are more explained by PC1. GNI and Edu2.FM are strongly correlated and are explained by PC2. Overall, the angles between arrows show correlations. The smaller the angle the stronger the correlation.
Part 4: personal interpretations of the first two principal component dimensions based on the biplot drawn after PCA on the standardized human data
Interpretation:
PC1 captures 53.6% of total variance in the original variables. PC2 captures 16.2% of variability. We see that Ado.Birth, Life.Exp, Edue.FM, Mat.Mor, and GNI have a small angle with PC1. This means that these variables have a high positive correlation with PC1. The same is true with PC2 and Labo.FM and Parli.F. The length of the arrows are proportional to the standard deviations of the features. We see that Mat.Mor, Labo.FM, and Life.Exp have the longest arrows. Variables pointing in the direction of PC1 are contributing to that PC1 dimension. The opposite is true for PC2.
Part 5: Load the tea dataset from the package Factominer
# code is from DataCamp
library(dplyr)
library(ggplot2)
library(tidyr)
library(FactoMineR)
data("tea")
dim(tea)## [1] 300 36str(tea)## 'data.frame': 300 obs. of 36 variables:
## $ breakfast : Factor w/ 2 levels "breakfast","Not.breakfast": 1 1 2 2 1 2 1 2 1 1 ...
## $ tea.time : Factor w/ 2 levels "Not.tea time",..: 1 1 2 1 1 1 2 2 2 1 ...
## $ evening : Factor w/ 2 levels "evening","Not.evening": 2 2 1 2 1 2 2 1 2 1 ...
## $ lunch : Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...
## $ dinner : Factor w/ 2 levels "dinner","Not.dinner": 2 2 1 1 2 1 2 2 2 2 ...
## $ always : Factor w/ 2 levels "always","Not.always": 2 2 2 2 1 2 2 2 2 2 ...
## $ home : Factor w/ 2 levels "home","Not.home": 1 1 1 1 1 1 1 1 1 1 ...
## $ work : Factor w/ 2 levels "Not.work","work": 1 1 2 1 1 1 1 1 1 1 ...
## $ tearoom : Factor w/ 2 levels "Not.tearoom",..: 1 1 1 1 1 1 1 1 1 2 ...
## $ friends : Factor w/ 2 levels "friends","Not.friends": 2 2 1 2 2 2 1 2 2 2 ...
## $ resto : Factor w/ 2 levels "Not.resto","resto": 1 1 2 1 1 1 1 1 1 1 ...
## $ pub : Factor w/ 2 levels "Not.pub","pub": 1 1 1 1 1 1 1 1 1 1 ...
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ sugar : Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ where : Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ price : Factor w/ 6 levels "p_branded","p_cheap",..: 4 6 6 6 6 3 6 6 5 5 ...
## $ age : int 39 45 47 23 48 21 37 36 40 37 ...
## $ sex : Factor w/ 2 levels "F","M": 2 1 1 2 2 2 2 1 2 2 ...
## $ SPC : Factor w/ 7 levels "employee","middle",..: 2 2 4 6 1 6 5 2 5 5 ...
## $ Sport : Factor w/ 2 levels "Not.sportsman",..: 2 2 2 1 2 2 2 2 2 1 ...
## $ age_Q : Factor w/ 5 levels "15-24","25-34",..: 3 4 4 1 4 1 3 3 3 3 ...
## $ frequency : Factor w/ 4 levels "1/day","1 to 2/week",..: 1 1 3 1 3 1 4 2 3 3 ...
## $ escape.exoticism: Factor w/ 2 levels "escape-exoticism",..: 2 1 2 1 1 2 2 2 2 2 ...
## $ spirituality : Factor w/ 2 levels "Not.spirituality",..: 1 1 1 2 2 1 1 1 1 1 ...
## $ healthy : Factor w/ 2 levels "healthy","Not.healthy": 1 1 1 1 2 1 1 1 2 1 ...
## $ diuretic : Factor w/ 2 levels "diuretic","Not.diuretic": 2 1 1 2 1 2 2 2 2 1 ...
## $ friendliness : Factor w/ 2 levels "friendliness",..: 2 2 1 2 1 2 2 1 2 1 ...
## $ iron.absorption : Factor w/ 2 levels "iron absorption",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ feminine : Factor w/ 2 levels "feminine","Not.feminine": 2 2 2 2 2 2 2 1 2 2 ...
## $ sophisticated : Factor w/ 2 levels "Not.sophisticated",..: 1 1 1 2 1 1 1 2 2 1 ...
## $ slimming : Factor w/ 2 levels "No.slimming",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ exciting : Factor w/ 2 levels "exciting","No.exciting": 2 1 2 2 2 2 2 2 2 2 ...
## $ relaxing : Factor w/ 2 levels "No.relaxing",..: 1 1 2 2 2 2 2 2 2 2 ...
## $ effect.on.health: Factor w/ 2 levels "effect on health",..: 2 2 2 2 2 2 2 2 2 2 ...# column names to keep in the dataset
keep_columns <- c("Tea", "How", "how", "sugar", "where", "lunch")
# select the 'keep_columns' to create a new dataset
tea_time <- select(tea, one_of(keep_columns))
# look at the summaries and structure of the data
summary(tea_time)## Tea How how sugar
## black : 74 alone:195 tea bag :170 No.sugar:155
## Earl Grey:193 lemon: 33 tea bag+unpackaged: 94 sugar :145
## green : 33 milk : 63 unpackaged : 36
## other: 9
## where lunch
## chain store :192 lunch : 44
## chain store+tea shop: 78 Not.lunch:256
## tea shop : 30
## str(tea_time)## 'data.frame': 300 obs. of 6 variables:
## $ Tea : Factor w/ 3 levels "black","Earl Grey",..: 1 1 2 2 2 2 2 1 2 1 ...
## $ How : Factor w/ 4 levels "alone","lemon",..: 1 3 1 1 1 1 1 3 3 1 ...
## $ how : Factor w/ 3 levels "tea bag","tea bag+unpackaged",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ sugar: Factor w/ 2 levels "No.sugar","sugar": 2 1 1 2 1 1 1 1 1 1 ...
## $ where: Factor w/ 3 levels "chain store",..: 1 1 1 1 1 1 1 1 2 2 ...
## $ lunch: Factor w/ 2 levels "lunch","Not.lunch": 2 2 2 2 2 2 2 2 2 2 ...# visualize the dataset
gather(tea_time) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free") + geom_bar() + theme(axis.text.x = element_text(angle = 45, hjust = 1, size = 8))
# multiple correspondence analysis
mca <- MCA(tea_time, graph = FALSE)
# summary of the model
summary(mca)##
## Call:
## MCA(X = tea_time, graph = FALSE)
##
##
## Eigenvalues
## Dim.1 Dim.2 Dim.3 Dim.4 Dim.5 Dim.6 Dim.7
## Variance 0.279 0.261 0.219 0.189 0.177 0.156 0.144
## % of var. 15.238 14.232 11.964 10.333 9.667 8.519 7.841
## Cumulative % of var. 15.238 29.471 41.435 51.768 61.434 69.953 77.794
## Dim.8 Dim.9 Dim.10 Dim.11
## Variance 0.141 0.117 0.087 0.062
## % of var. 7.705 6.392 4.724 3.385
## Cumulative % of var. 85.500 91.891 96.615 100.000
##
## Individuals (the 10 first)
## Dim.1 ctr cos2 Dim.2 ctr cos2 Dim.3
## 1 | -0.298 0.106 0.086 | -0.328 0.137 0.105 | -0.327
## 2 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 3 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 4 | -0.530 0.335 0.460 | -0.318 0.129 0.166 | 0.211
## 5 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 6 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 7 | -0.369 0.162 0.231 | -0.300 0.115 0.153 | -0.202
## 8 | -0.237 0.067 0.036 | -0.136 0.024 0.012 | -0.695
## 9 | 0.143 0.024 0.012 | 0.871 0.969 0.435 | -0.067
## 10 | 0.476 0.271 0.140 | 0.687 0.604 0.291 | -0.650
## ctr cos2
## 1 0.163 0.104 |
## 2 0.735 0.314 |
## 3 0.062 0.069 |
## 4 0.068 0.073 |
## 5 0.062 0.069 |
## 6 0.062 0.069 |
## 7 0.062 0.069 |
## 8 0.735 0.314 |
## 9 0.007 0.003 |
## 10 0.643 0.261 |
##
## Categories (the 10 first)
## Dim.1 ctr cos2 v.test Dim.2 ctr cos2
## black | 0.473 3.288 0.073 4.677 | 0.094 0.139 0.003
## Earl Grey | -0.264 2.680 0.126 -6.137 | 0.123 0.626 0.027
## green | 0.486 1.547 0.029 2.952 | -0.933 6.111 0.107
## alone | -0.018 0.012 0.001 -0.418 | -0.262 2.841 0.127
## lemon | 0.669 2.938 0.055 4.068 | 0.531 1.979 0.035
## milk | -0.337 1.420 0.030 -3.002 | 0.272 0.990 0.020
## other | 0.288 0.148 0.003 0.876 | 1.820 6.347 0.102
## tea bag | -0.608 12.499 0.483 -12.023 | -0.351 4.459 0.161
## tea bag+unpackaged | 0.350 2.289 0.056 4.088 | 1.024 20.968 0.478
## unpackaged | 1.958 27.432 0.523 12.499 | -1.015 7.898 0.141
## v.test Dim.3 ctr cos2 v.test
## black 0.929 | -1.081 21.888 0.382 -10.692 |
## Earl Grey 2.867 | 0.433 9.160 0.338 10.053 |
## green -5.669 | -0.108 0.098 0.001 -0.659 |
## alone -6.164 | -0.113 0.627 0.024 -2.655 |
## lemon 3.226 | 1.329 14.771 0.218 8.081 |
## milk 2.422 | 0.013 0.003 0.000 0.116 |
## other 5.534 | -2.524 14.526 0.197 -7.676 |
## tea bag -6.941 | -0.065 0.183 0.006 -1.287 |
## tea bag+unpackaged 11.956 | 0.019 0.009 0.000 0.226 |
## unpackaged -6.482 | 0.257 0.602 0.009 1.640 |
##
## Categorical variables (eta2)
## Dim.1 Dim.2 Dim.3
## Tea | 0.126 0.108 0.410 |
## How | 0.076 0.190 0.394 |
## how | 0.708 0.522 0.010 |
## sugar | 0.065 0.001 0.336 |
## where | 0.702 0.681 0.055 |
## lunch | 0.000 0.064 0.111 |# visualize MCA
plot(mca, invisible=c("ind"), habillage = "quali", graph.type = "classic")
Interpretation:
The Tea dataset relates to the consumption of tea and how, when, and where it is consumed. There are 300 observations and 36 variables. We see that Earl Grey is the favority type of tea and that it is mainly consumed alone. The tea is mainly bought in chain store and consumed before or after lunch (not much during lunch). The tea is from tea bags and there is an equal use or not of sugar with it. We see that milk, Earl Grey, sugar, tea bags, and chain store are in the direction of Dim2. This group is in the middle left side of the plot. The other features are more dispersed and are in the right side of the plot. Since these features are closer to the origin, we can say that more people use them. For instance, Earl Grey vs black tea. Early Grey is closer to the origin and therefore, more people use Early Grey than black tea. This is more so with milk and lemon.
Chapter 6: Analysis of longitudinal data
Part 1: Implement the analyses of Chapter 8 of MABS
RATS dataset
# most of the code is from DataCamp
# set plots size
knitr::opts_chunk$set(fig.width=16, fig.height=10)
# Loading RATS dataset
# libraries
library(dplyr)
library(tidyr)
# load rats data
RATS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/rats.txt", sep = "\t", header = TRUE)
# Factor variables ID and Group
RATS$ID <- factor(RATS$ID)
RATS$Group <- factor(RATS$Group)
RATS## ID Group WD1 WD8 WD15 WD22 WD29 WD36 WD43 WD44 WD50 WD57 WD64
## 1 1 1 240 250 255 260 262 258 266 266 265 272 278
## 2 2 1 225 230 230 232 240 240 243 244 238 247 245
## 3 3 1 245 250 250 255 262 265 267 267 264 268 269
## 4 4 1 260 255 255 265 265 268 270 272 274 273 275
## 5 5 1 255 260 255 270 270 273 274 273 276 278 280
## 6 6 1 260 265 270 275 275 277 278 278 284 279 281
## 7 7 1 275 275 260 270 273 274 276 271 282 281 284
## 8 8 1 245 255 260 268 270 265 265 267 273 274 278
## 9 9 2 410 415 425 428 438 443 442 446 456 468 478
## 10 10 2 405 420 430 440 448 460 458 464 475 484 496
## 11 11 2 445 445 450 452 455 455 451 450 462 466 472
## 12 12 2 555 560 565 580 590 597 595 595 612 618 628
## 13 13 3 470 465 475 485 487 493 493 504 507 518 525
## 14 14 3 535 525 530 533 535 540 525 530 543 544 559
## 15 15 3 520 525 530 540 543 546 538 544 553 555 548
## 16 16 3 510 510 520 515 530 538 535 542 550 553 569# Convert data to long form
RATSL <- RATS %>%
gather(key = WD, value = Weight, -ID, -Group) %>%
mutate(Day = as.integer(substr(WD,3,4)))
# dataset
print(RATSL, row.names = FALSE)## ID Group WD Weight Day
## 1 1 WD1 240 1
## 2 1 WD1 225 1
## 3 1 WD1 245 1
## 4 1 WD1 260 1
## 5 1 WD1 255 1
## 6 1 WD1 260 1
## 7 1 WD1 275 1
## 8 1 WD1 245 1
## 9 2 WD1 410 1
## 10 2 WD1 405 1
## 11 2 WD1 445 1
## 12 2 WD1 555 1
## 13 3 WD1 470 1
## 14 3 WD1 535 1
## 15 3 WD1 520 1
## 16 3 WD1 510 1
## 1 1 WD8 250 8
## 2 1 WD8 230 8
## 3 1 WD8 250 8
## 4 1 WD8 255 8
## 5 1 WD8 260 8
## 6 1 WD8 265 8
## 7 1 WD8 275 8
## 8 1 WD8 255 8
## 9 2 WD8 415 8
## 10 2 WD8 420 8
## 11 2 WD8 445 8
## 12 2 WD8 560 8
## 13 3 WD8 465 8
## 14 3 WD8 525 8
## 15 3 WD8 525 8
## 16 3 WD8 510 8
## 1 1 WD15 255 15
## 2 1 WD15 230 15
## 3 1 WD15 250 15
## 4 1 WD15 255 15
## 5 1 WD15 255 15
## 6 1 WD15 270 15
## 7 1 WD15 260 15
## 8 1 WD15 260 15
## 9 2 WD15 425 15
## 10 2 WD15 430 15
## 11 2 WD15 450 15
## 12 2 WD15 565 15
## 13 3 WD15 475 15
## 14 3 WD15 530 15
## 15 3 WD15 530 15
## 16 3 WD15 520 15
## 1 1 WD22 260 22
## 2 1 WD22 232 22
## 3 1 WD22 255 22
## 4 1 WD22 265 22
## 5 1 WD22 270 22
## 6 1 WD22 275 22
## 7 1 WD22 270 22
## 8 1 WD22 268 22
## 9 2 WD22 428 22
## 10 2 WD22 440 22
## 11 2 WD22 452 22
## 12 2 WD22 580 22
## 13 3 WD22 485 22
## 14 3 WD22 533 22
## 15 3 WD22 540 22
## 16 3 WD22 515 22
## 1 1 WD29 262 29
## 2 1 WD29 240 29
## 3 1 WD29 262 29
## 4 1 WD29 265 29
## 5 1 WD29 270 29
## 6 1 WD29 275 29
## 7 1 WD29 273 29
## 8 1 WD29 270 29
## 9 2 WD29 438 29
## 10 2 WD29 448 29
## 11 2 WD29 455 29
## 12 2 WD29 590 29
## 13 3 WD29 487 29
## 14 3 WD29 535 29
## 15 3 WD29 543 29
## 16 3 WD29 530 29
## 1 1 WD36 258 36
## 2 1 WD36 240 36
## 3 1 WD36 265 36
## 4 1 WD36 268 36
## 5 1 WD36 273 36
## 6 1 WD36 277 36
## 7 1 WD36 274 36
## 8 1 WD36 265 36
## 9 2 WD36 443 36
## 10 2 WD36 460 36
## 11 2 WD36 455 36
## 12 2 WD36 597 36
## 13 3 WD36 493 36
## 14 3 WD36 540 36
## 15 3 WD36 546 36
## 16 3 WD36 538 36
## 1 1 WD43 266 43
## 2 1 WD43 243 43
## 3 1 WD43 267 43
## 4 1 WD43 270 43
## 5 1 WD43 274 43
## 6 1 WD43 278 43
## 7 1 WD43 276 43
## 8 1 WD43 265 43
## 9 2 WD43 442 43
## 10 2 WD43 458 43
## 11 2 WD43 451 43
## 12 2 WD43 595 43
## 13 3 WD43 493 43
## 14 3 WD43 525 43
## 15 3 WD43 538 43
## 16 3 WD43 535 43
## 1 1 WD44 266 44
## 2 1 WD44 244 44
## 3 1 WD44 267 44
## 4 1 WD44 272 44
## 5 1 WD44 273 44
## 6 1 WD44 278 44
## 7 1 WD44 271 44
## 8 1 WD44 267 44
## 9 2 WD44 446 44
## 10 2 WD44 464 44
## 11 2 WD44 450 44
## 12 2 WD44 595 44
## 13 3 WD44 504 44
## 14 3 WD44 530 44
## 15 3 WD44 544 44
## 16 3 WD44 542 44
## 1 1 WD50 265 50
## 2 1 WD50 238 50
## 3 1 WD50 264 50
## 4 1 WD50 274 50
## 5 1 WD50 276 50
## 6 1 WD50 284 50
## 7 1 WD50 282 50
## 8 1 WD50 273 50
## 9 2 WD50 456 50
## 10 2 WD50 475 50
## 11 2 WD50 462 50
## 12 2 WD50 612 50
## 13 3 WD50 507 50
## 14 3 WD50 543 50
## 15 3 WD50 553 50
## 16 3 WD50 550 50
## 1 1 WD57 272 57
## 2 1 WD57 247 57
## 3 1 WD57 268 57
## 4 1 WD57 273 57
## 5 1 WD57 278 57
## 6 1 WD57 279 57
## 7 1 WD57 281 57
## 8 1 WD57 274 57
## 9 2 WD57 468 57
## 10 2 WD57 484 57
## 11 2 WD57 466 57
## 12 2 WD57 618 57
## 13 3 WD57 518 57
## 14 3 WD57 544 57
## 15 3 WD57 555 57
## 16 3 WD57 553 57
## 1 1 WD64 278 64
## 2 1 WD64 245 64
## 3 1 WD64 269 64
## 4 1 WD64 275 64
## 5 1 WD64 280 64
## 6 1 WD64 281 64
## 7 1 WD64 284 64
## 8 1 WD64 278 64
## 9 2 WD64 478 64
## 10 2 WD64 496 64
## 11 2 WD64 472 64
## 12 2 WD64 628 64
## 13 3 WD64 525 64
## 14 3 WD64 559 64
## 15 3 WD64 548 64
## 16 3 WD64 569 64Plot of the RATSL data
#Access the package ggplot2
library(ggplot2)
# Glimpse the data
glimpse(RATSL)## Rows: 176
## Columns: 5
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3,~
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 1, ~
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", ~
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, 470~
## $ Day <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, ~# Draw the plot
ggplot(RATSL, aes(x = Day, y = Weight, linetype = ID, color = ID)) +
geom_line() + geom_point() +
facet_grid(. ~ Group, labeller = label_both)
From the above plots, we see that each group of rats started with a different initial weight. Group 1 gained the least amount of weight during the study period. Group 2 and group 3 did gain weight at approximately the same rate.
Plots of the RATSL data after standartization
# Standardise the variable RATS
RATSL <- RATSL %>%
group_by(Day) %>%
mutate(stdweight = (Weight - mean(Weight))/sd(Weight) ) %>%
ungroup()
# Glimpse the data
glimpse(RATSL)## Rows: 176
## Columns: 6
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2,~
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, ~
## $ WD <chr> "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1", "WD1~
## $ Weight <int> 240, 225, 245, 260, 255, 260, 275, 245, 410, 405, 445, 555, ~
## $ Day <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, ~
## $ stdweight <dbl> -1.0011429, -1.1203857, -0.9613953, -0.8421525, -0.8819001, ~# Plot again with the standardised RATS
ggplot(RATSL, aes(x = Day, y = stdweight, linetype = ID, color = ID)) +
geom_line() + geom_point() +
scale_linetype_manual(values = rep(1:16, Days=4)) +
facet_grid(. ~ Group, labeller = label_both) +
scale_y_continuous(name = "standardized Weight")
From the above standardized plots, we see that group 1 did not gain or lose weight. We also see that group 2 gained the most weight and group 3 lost weight.
Summary Graph
# Number of weeks, baseline (week 0) included
n <- 9
# Summary data with mean and standard error of RATS by Group and Day
RATSS <- RATSL %>%
group_by(Group, Day) %>%
summarise( mean = mean(Weight), se = sd(Weight)/sqrt(n) ) %>%
ungroup()
# Glimpse the data
glimpse(RATSS)## Rows: 33
## Columns: 4
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2~
## $ Day <int> 1, 8, 15, 22, 29, 36, 43, 44, 50, 57, 64, 1, 8, 15, 22, 29, 36, ~
## $ mean <dbl> 250.625, 255.000, 254.375, 261.875, 264.625, 265.000, 267.375, 2~
## $ se <dbl> 5.073859, 4.364358, 3.825302, 4.533605, 3.685827, 3.927922, 3.65~# Plot the mean profiles
ggplot(RATSS, aes(x = Day, y = mean, linetype = Group, shape = Group, color = Group)) +
geom_line() +
scale_linetype_manual(values = c(1,2,3)) +
geom_point(size=3) +
scale_shape_manual(values = c(1,2,3)) +
geom_errorbar(aes(ymin = mean - se, ymax = mean + se, linetype="1"), width=0.3) +
theme(legend.position = c(0.8,0.8)) +
scale_y_continuous(name = "mean(Weight) +/- se(Weight)")
From the summary plot with mean and standard error, we see that group 1 did not gain much weight during the study but group 2 and group 3 did gain weight at a similar rate.
Outliners
# Create a summary data
RATSL8S <- RATSL %>%
filter(Day > 0) %>%
group_by(Group, ID) %>%
summarise( mean=mean(Weight) ) %>%
ungroup()
# Glimpse the data
glimpse(RATSL8S)## Rows: 16
## Columns: 3
## $ Group <fct> 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3
## $ ID <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16
## $ mean <dbl> 261.0909, 237.6364, 260.1818, 266.5455, 269.4545, 274.7273, 274.~# Draw a boxplot of the mean versus treatment
ggplot(RATSL8S, aes(x = Group, y = mean, color = Group)) +
geom_boxplot(outlier.colour="blue", outlier.shape=16,
outlier.size=10, notch=FALSE) +
stat_summary(fun = mean, geom = "point", shape=23, size=10, fill = "red", colour = "blue") +
scale_y_continuous(name = "mean(Weight), weeks 1-9")
From the above boxplot, group 2 seems to have more variation than group 1 and 3. As well, we see 3 outliers.
# Add the baseline from the original data as a new variable to the summary data
RATSL8S2 <- RATSL8S %>%
mutate(baseline = RATS$WD1)
# Fit the linear model with the mean as the response
fit <- lm(mean ~ baseline + Group, data = RATSL8S2)
# Compute the analysis of variance table for the fitted model with anova()
anova(fit)## Analysis of Variance Table
##
## Response: mean
## Df Sum Sq Mean Sq F value Pr(>F)
## baseline 1 252125 252125 2237.0655 5.217e-15 ***
## Group 2 726 363 3.2219 0.07586 .
## Residuals 12 1352 113
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Given that Pr(>F) is smaller than 0.05, we reject the null hypothesis (all means are equal) and we conclude that at least one group is different in weight than the others.
Part 2: Implement the analyses of Chapter 9 of MABS
# load BPRS data
BPRS <- read.table("https://raw.githubusercontent.com/KimmoVehkalahti/MABS/master/Examples/data/BPRS.txt", sep = " ", header = TRUE)
# Data
BPRS## treatment subject week0 week1 week2 week3 week4 week5 week6 week7 week8
## 1 1 1 42 36 36 43 41 40 38 47 51
## 2 1 2 58 68 61 55 43 34 28 28 28
## 3 1 3 54 55 41 38 43 28 29 25 24
## 4 1 4 55 77 49 54 56 50 47 42 46
## 5 1 5 72 75 72 65 50 39 32 38 32
## 6 1 6 48 43 41 38 36 29 33 27 25
## 7 1 7 71 61 47 30 27 40 30 31 31
## 8 1 8 30 36 38 38 31 26 26 25 24
## 9 1 9 41 43 39 35 28 22 20 23 21
## 10 1 10 57 51 51 55 53 43 43 39 32
## 11 1 11 30 34 34 41 36 36 38 36 36
## 12 1 12 55 52 49 54 48 43 37 36 31
## 13 1 13 36 32 36 31 25 25 21 19 22
## 14 1 14 38 35 36 34 25 27 25 26 26
## 15 1 15 66 68 65 49 36 32 27 30 37
## 16 1 16 41 35 45 42 31 31 29 26 30
## 17 1 17 45 38 46 38 40 33 27 31 27
## 18 1 18 39 35 27 25 29 28 21 25 20
## 19 1 19 24 28 31 28 29 21 22 23 22
## 20 1 20 38 34 27 25 25 27 21 19 21
## 21 2 1 52 73 42 41 39 38 43 62 50
## 22 2 2 30 23 32 24 20 20 19 18 20
## 23 2 3 65 31 33 28 22 25 24 31 32
## 24 2 4 37 31 27 31 31 26 24 26 23
## 25 2 5 59 67 58 61 49 38 37 36 35
## 26 2 6 30 33 37 33 28 26 27 23 21
## 27 2 7 69 52 41 33 34 37 37 38 35
## 28 2 8 62 54 49 39 55 51 55 59 66
## 29 2 9 38 40 38 27 31 24 22 21 21
## 30 2 10 65 44 31 34 39 34 41 42 39
## 31 2 11 78 95 75 76 66 64 64 60 75
## 32 2 12 38 41 36 27 29 27 21 22 23
## 33 2 13 63 65 60 53 52 32 37 52 28
## 34 2 14 40 37 31 38 35 30 33 30 27
## 35 2 15 40 36 55 55 42 30 26 30 37
## 36 2 16 54 45 35 27 25 22 22 22 22
## 37 2 17 33 41 30 32 46 43 43 43 43
## 38 2 18 28 30 29 33 30 26 36 33 30
## 39 2 19 52 43 26 27 24 32 21 21 21
## 40 2 20 47 36 32 29 25 23 23 23 23# Factor treatment & subject
BPRS$treatment <- factor(BPRS$treatment)
BPRS$subject <- factor(BPRS$subject)
# Take a glimpse at the BPRSL data
glimpse(BPRS)## Rows: 40
## Columns: 11
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ~
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1~
## $ week0 <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, 38, 66, ~
## $ week1 <int> 36, 68, 55, 77, 75, 43, 61, 36, 43, 51, 34, 52, 32, 35, 68, ~
## $ week2 <int> 36, 61, 41, 49, 72, 41, 47, 38, 39, 51, 34, 49, 36, 36, 65, ~
## $ week3 <int> 43, 55, 38, 54, 65, 38, 30, 38, 35, 55, 41, 54, 31, 34, 49, ~
## $ week4 <int> 41, 43, 43, 56, 50, 36, 27, 31, 28, 53, 36, 48, 25, 25, 36, ~
## $ week5 <int> 40, 34, 28, 50, 39, 29, 40, 26, 22, 43, 36, 43, 25, 27, 32, ~
## $ week6 <int> 38, 28, 29, 47, 32, 33, 30, 26, 20, 43, 38, 37, 21, 25, 27, ~
## $ week7 <int> 47, 28, 25, 42, 38, 27, 31, 25, 23, 39, 36, 36, 19, 26, 30, ~
## $ week8 <int> 51, 28, 24, 46, 32, 25, 31, 24, 21, 32, 36, 31, 22, 26, 37, ~# Convert to long form
BPRSL <- BPRS %>% gather(key = weeks, value = bprs, -treatment, -subject)
# Extract the week number
BPRSL <- BPRSL %>% mutate(week = as.integer(substr(weeks,5,5)))
glimpse(BPRSL)## Rows: 360
## Columns: 5
## $ treatment <fct> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ~
## $ subject <fct> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1~
## $ weeks <chr> "week0", "week0", "week0", "week0", "week0", "week0", "week0~
## $ bprs <int> 42, 58, 54, 55, 72, 48, 71, 30, 41, 57, 30, 55, 36, 38, 66, ~
## $ week <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ~First Plot
# Check the dimensions of the data
dim(BPRSL)## [1] 360 5# Draw the plot
ggplot(BPRSL, aes(x = week, y = bprs, linetype = subject)) +
geom_line() +
scale_linetype_manual(values = rep(1:10, times=4)) +
facet_grid(. ~ treatment, labeller = label_both) +
theme(legend.position = "none") +
scale_y_continuous(limits = c(min(BPRSL$bprs), max(BPRSL$bprs)))
From the first plot, we see that there is a downward trend for some participants in treatment 1 and 2. However, for some participants, bprs is increasing in treatment 1 and 2.
The Linear model
# create a regression model BPRS_reg
BPRS_reg <- lm(bprs ~ week + treatment, data = BPRSL)
# print out a summary of the model
summary(BPRS_reg)##
## Call:
## lm(formula = bprs ~ week + treatment, data = BPRSL)
##
## Residuals:
## Min 1Q Median 3Q Max
## -22.454 -8.965 -3.196 7.002 50.244
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 46.4539 1.3670 33.982 <2e-16 ***
## week -2.2704 0.2524 -8.995 <2e-16 ***
## treatment2 0.5722 1.3034 0.439 0.661
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.37 on 357 degrees of freedom
## Multiple R-squared: 0.1851, Adjusted R-squared: 0.1806
## F-statistic: 40.55 on 2 and 357 DF, p-value: < 2.2e-16Here, only week as some significance but treatment2 does not.
The Random Intercept Model
# access library lme4
library(lme4)
# Create a random intercept model
BPRS_ref <- lmer(bprs ~ week + treatment + (1 | subject), data = BPRSL, REML = FALSE)
# Print the summary of the model
summary(BPRS_ref)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (1 | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2748.7 2768.1 -1369.4 2738.7 355
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0481 -0.6749 -0.1361 0.4813 3.4855
##
## Random effects:
## Groups Name Variance Std.Dev.
## subject (Intercept) 47.41 6.885
## Residual 104.21 10.208
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 1.9090 24.334
## week -2.2704 0.2084 -10.896
## treatment2 0.5722 1.0761 0.532
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.437
## treatment2 -0.282 0.000Fitted <- fitted(BPRS_ref)
ggplot(BPRSL, aes(x = week, y = Fitted, group = subject)) +
geom_line() +
scale_x_continuous(name = "week") +
scale_y_continuous(limits = c(min(BPRSL$bprs), max(BPRSL$bprs))) +
theme(legend.position = "top") +
labs(title = "The Random Intercept Model")
Random Intercept and Random Slope Model
# create a random intercept and random slope model
BPRS_ref1 <- lmer(bprs ~ week + treatment + (week | subject), data = BPRSL, REML = FALSE)
# print a summary of the model
summary(BPRS_ref1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week + treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2745.4 2772.6 -1365.7 2731.4 353
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8919 -0.6194 -0.0691 0.5531 3.7976
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.8222 8.0512
## week 0.9609 0.9802 -0.51
## Residual 97.4305 9.8707
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 46.4539 2.1052 22.066
## week -2.2704 0.2977 -7.626
## treatment2 0.5722 1.0405 0.550
##
## Correlation of Fixed Effects:
## (Intr) week
## week -0.582
## treatment2 -0.247 0.000# perform an ANOVA test on the two models
anova(BPRS_ref1, BPRS_ref)## Data: BPRSL
## Models:
## BPRS_ref: bprs ~ week + treatment + (1 | subject)
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## BPRS_ref 5 2748.7 2768.1 -1369.4 2738.7
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4 7.2721 2 0.02636 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Fitted <- fitted(BPRS_ref1)
ggplot(BPRSL, aes(x = week, y = Fitted, group = subject)) +
geom_line() +
scale_x_continuous(name = "week") +
scale_y_continuous(name = "BPRS") +
theme(legend.position = "top") +
labs(title = "Random Intercept and Random Slope Model")
Here, we see a better fit against the comparison model for BPRS_ref1 with a chi-square of 0.02636.
Random Intercept and Random Slope Model with interaction
# create a random intercept and random slope model
BPRS_ref2 <- lmer(bprs ~ week * treatment + (week | subject), data = BPRSL, REML = FALSE)
# print a summary of the model
summary(BPRS_ref2)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: bprs ~ week * treatment + (week | subject)
## Data: BPRSL
##
## AIC BIC logLik deviance df.resid
## 2744.3 2775.4 -1364.1 2728.3 352
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.0512 -0.6271 -0.0768 0.5288 3.9260
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## subject (Intercept) 64.9964 8.0620
## week 0.9687 0.9842 -0.51
## Residual 96.4707 9.8220
## Number of obs: 360, groups: subject, 20
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 47.8856 2.2521 21.262
## week -2.6283 0.3589 -7.323
## treatment2 -2.2911 1.9090 -1.200
## week:treatment2 0.7158 0.4010 1.785
##
## Correlation of Fixed Effects:
## (Intr) week trtmn2
## week -0.650
## treatment2 -0.424 0.469
## wek:trtmnt2 0.356 -0.559 -0.840# perform an ANOVA test on the two models
anova(BPRS_ref2, BPRS_ref1)## Data: BPRSL
## Models:
## BPRS_ref1: bprs ~ week + treatment + (week | subject)
## BPRS_ref2: bprs ~ week * treatment + (week | subject)
## npar AIC BIC logLik deviance Chisq Df Pr(>Chisq)
## BPRS_ref1 7 2745.4 2772.6 -1365.7 2731.4
## BPRS_ref2 8 2744.3 2775.4 -1364.1 2728.3 3.1712 1 0.07495 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Create a vector of the fitted values
Fitted <- fitted(BPRS_ref2)
# Create a new column fitted to BPRSL
BPRSL2 <- BPRSL %>%
mutate(Fitted)
ggplot(BPRSL2, aes(x = week, y = Fitted, group = subject)) +
geom_line() +
scale_x_continuous(name = "week") +
scale_y_continuous(name = "BPRS") +
theme(legend.position = "top") +
labs(title = "Random Intercept and Random Slope Model with interaction")
Here, we don’t see a better fit so, BPRS_ref1 has the best fit.
Overall, we see a decrease in bprs for all participants.